Symmetry and reversibility of martensitic

transformations

Kaushik Bhattacharya?§, Sergio Conti†§,Giovanni Zanzotto‡§ and Johannes Zimmer†§

December 23, 2003

? Division of Engineering and Applied Science,California Institute of Technology, Pasadena CA 91125, USA

† Max Planck Institute for Mathematics in the Sciences,Inselstr. 22, 04103 Leipzig, Germany

‡ Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate,Universita di Padova, Via Belzoni 7, 35131 Padova, Italy

§ All authors contributed equally to this work.

Martensitic transformations are diffusionless solid-to-solid phasetransformations characterized by a rapid change of crystal structure,observed in metals, alloys, ceramics, and proteins1,2. They come intwo widely different classes. In steels, the transformation microstruc-ture induced by quenching remains essentially unchanged upon sub-sequent loading or heating; the transformation is not reversible. Inshape-memory alloys, on the contrary, the microstructures formed oncooling are easily manipulated by loads and disappear upon reheat-ing; the transformation is reversible. Here we show that these sharpdifferences are dictated by the symmetry of the energetic landscape.In weak transformations the symmetry groups of both phases are in-cluded in a common finite symmetry group3, in reconstructive trans-formations they are not4. We demonstrate that for reconstructivetransformations the energy barrier to lattice-invariant shears (as intwinning or slip) is no higher than the barrier to the phase transitionitself. A remarkable implication is that reconstructive transforma-tions are accompanied by plastic deformation through dislocationsand twinning in the parent phase, making these phase changes irre-versible. In contrast, for weak transformations, the energy barrierto lattice-invariant shears is independent of that to the phase tran-sition. Consequently, weak transformations can occur with virtuallyno plasticity and are potentially reversible.

Martensitic transformations are at the basis of numerous technological ap-plications. Most notable amongst these is in steel, where the transforma-tion induced by quenching (fast cooling) is exploited for enhancing the alloy’sstrength1. Another is the fascinating shape-memory effect in alloys like Nitinol,used in medical and engineering devices5. Martensitic phase changes are alsoexploited to toughen structural ceramics6 such as zirconia, and observed in bi-ological systems such as the tail sheath of the T4 bacteriophage virus7. Ideas

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originating from the study of these transformations have led to improved materi-als for actuation (ferromagnetic shape-memory alloys8,9 and ferroelectrics10,11)and to candidates for artificial muscles12. Finally, the rich microstructure (dis-tinctive patterns developed at scales ranging from a few nanometers to a fewmicrons) that accompanies these transitions, has made this a valuable theoret-ical sand-box for the development of multi-scale modeling tools13.

In some materials such as shape-memory alloys, the martensitic phase changeis almost perfectly reversible (and often termed ‘thermoelastic’). As one startsat high temperature and cools a shape-memory alloy like Nitinol, the trans-formation proceeds by the appearance and growth of microstructure with a(twinned) plate-like morphology; the transformation from start to finish takesabout 20 degrees Celsius. The microstructure is mobile and can be changed bythe application of loads. The transformation can be completely reversed, withthe disappearance of the microstructure upon reheating and the appearanceof little or no dislocations or twinning in the parent (high temperature, highsymmetry) phase.

In stark contrast, the martensitic transformation is not reversible in materi-als like steel and other alloys, such as CoNi. In steels, the microstructure formsin a sudden burst with a lath-like morphology upon cooling, and is immobileon loading. Moreover, the transformation is irreversible and the microstructuredoes not disappear upon reheating. In CoNi the phase change is also irreversible,as successive microstructures form both on heating and cooling. These materialsundergoing irreversible transformations are characterized by significant disloca-tions and twinning in the parent phase.

We provide an explanation for this difference in (ir-)reversibility on the basisof the symmetry change during the transformation. We call weak the martensitictransformations in which the symmetry group of both the parent and productphase are included in a common finite symmetry group3 (which includes symme-try breaking), and reconstructive otherwise4 (note that this is different from theusage of the term ‘reconstructive’ to mean ‘diffusional’; all the transformationsconsidered here are diffusionless). We show through rigorous mathematical the-ory and numerical simulation that irreversibility is inevitable in a reconstructivephase transformation, and not so in a weak transition.

Fig. 1 illustrates our main idea through a square-to-hexagonal reconstructivephase change in a two-dimensional crystal. Consider first a square lattice with aunit cell shown on the left, and suppose this cell is transformed to the unit cell ofthe hexagonal lattice shown in solid (middle). The symmetry of the hexagonallattice implies that the solid and dashed unit cells shown in the middle areequivalent. If the crystal is transformed back to the square phase, the dashedhexagonal cell can go to, say, the dashed cell on the right. Crucially, the squarecell on the left is then transformed to the sheared cell of the square lattice onthe right. In short, upon transforming, performing a symmetry operation, andtransforming back, we have deformed the crystal through a lattice-invariantshear, i.e., a shearing deformation that leaves the entire (ideal, infinite) latticeinvariant. Our results hold for finite lattices if boundary effects can be neglected,as illustrated below by means of numerical simulations.

Various experimental observations confirm our predictions. Pure iron (Fe)undergoes a temperature induced reconstructive γ-α (face-centered-cubic tobody-centered-cubic, fcc-to-bcc) transformation. As Ni and C are added toobtain steel, the martensite becomes body-centred tetragonal (bct) with an in-

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Figure 1. A lattice-invariant shear can be generated by a forward and reversesquare-to-hexagonal phase transition. The transformation takes the solid squareon the left to the solid rhombus in the middle. The hexagonal symmetry impliesthe equivalence of the solid rhombic cell to the dashed rhombic one. The reversetransformation takes the latter to the dashed square on the right. In the process,the original solid square on the left has sheared by a lattice-invariant shear tothe solid parallelogram on the right.

creasing tetragonality, so that the martensitic transformation becomes weak.Correspondingly, Maki and Tamura14 found that the reversibility of the phasechanges increased, the amount of plastic deformation decreased, and the mor-phology changed from lath to butterfly to lenticular to plate-like with increasingNi and C. Iron also undergoes a pressure-induced reconstructive α-ε (bcc tohexagonal-close-packed, hcp) transformation, again accompanied by both twinsand dislocations15,16. Another set of observations concern materials undergoingthe reconstructive fcc-to-hcp transformation. Liu et al.17 considered the CoNisystem; they hardened the parent phase to prevent any plastic deformation,but found that the transition was still irreversible, with both heating and cool-ing producing twins and plastic deformation. Similar observations were madein FeMnCrSiNi steels18,19. Finally, in a block co-polymer, the (reconstructive)bcc-to-hexagonal transformation was observed to be irreversible because of ori-entation proliferation through twinning in both phases20.

In spirit, our results are related to those of Otsuka and Shimizu21 who discussthe effects of ordering on crystallographic reversibility of martensitic transfor-mation in alloys. They observe that in ordered alloys the transformation pathmust be such that the order is not destroyed, whereas in disordered ones onlythe atomic positions need to be recovered. While their results are consistentwith many experiments, they also note that the fcc-to-face-centered-tetragonal(fct) transformation is an ‘exception’, which they argue is a consequence of thefact that ‘the lattice correspondence is unique in the reverse transformation be-cause of the so simple lattice change and lower symmetry of the fct phase’. Weshow here that the essential difference depends on the crystal symmetry, ratherthan on order and disorder. This provides a general framework which includesboth their theory and exception, and also generalizes to other situations.

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The common explanation (e.g., Maki and Tamura14) for the irreversibilityin Fe (or low Ni or C steels) is based on the volume change that accompaniesthe transition. The idea is that a partially transformed region causes stressand plastic deformation. However, a system like the styrene block co-polymerconsidered by Lee et al.20 should largely be unaffected by any volume change.Our explanation is independent of such a volume effect. It is possible, however,that both mechanisms contribute to the irreversibility in Fe.

As our result is only based on symmetry, it is independent of material pa-rameters, such as the elastic moduli. We show that lattice-invariant shears inreconstructive transformations cost only as much as the phase transition it-self, making reconstructive transformations irreversible. On the other hand,weak transformations have the potential to be reversible, because the energybarriers to lattice-invariant shears and the phase transition are independentof each other. Our numerical simulations indicate that they are in fact re-versible in a generic situation. However, the barriers might happen to be com-parable in particular materials undergoing a weak transformation. This forexample is the case in Ni-50Ti, and plastic deformation masks the reversibletransformation. However, (precipitate) hardening this material makes thesebarriers different thereby revealing the reversible character of the underlyingweak transition5. In contrast, hardening does not rescue the irreversibility ofCoNi17 or FeMnCrSiNi18,19, which undergo reconstructive transformations. Insummary, being weak is a necessary condition for reversibility.

Another consequence of our remarks is that materials like FeNi should becomparatively softer at compositions that are closer to a reconstructive trans-formation. This effect, however, may be obscured by the fact that the verysoftness of the ideal lattice produces plastic deformation and the entanglementof dislocations in the crystal, leading eventually to hardening. None of thesephenomena are present in crystals undergoing weak martensitic transformations.

We now present the general theory followed by a concrete example, andnumerical simulations. We state the theory for the simple case of Bravais lat-tices, which can be readily extended to crystals whose translational symmetryis not discontinuous across the phase change. A Bravais lattice L(ei) is given bythe linear combinations, with integral coefficients, of three independent vectors{e1, e2, e3} forming the lattice basis. Another basis {f1, f2, f3} generates thesame lattice, L(ei) = L(fi), if and only if fi =

∑3j=1 mj

iej with a matrix m

belonging to the group22,3,23

G := {m : mji integers and det(m) = ±1}, (1)

which is the global symmetry group of Bravais lattices. Applied to any givenlattice, G consists of rotations, reflections, lattice-invariant shears, and combi-nations thereof; the restriction to rotations and reflections gives the lattice group(a matrix representation of the point group) of that lattice.

The free energy density Φ of the crystal is a function of the lattice basis ata fixed temperature. It is invariant under the global symmetry group G, as itcannot distinguish among bases generating the same lattice24,23,13:

Φ(ei) = Φ(mjiej) for every m ∈ G. (2)

This global framework takes all possible deformations into account, includinglarge shearing distortions. Due to (2) the energy landscape of the crystal has

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infinitely many wells, which are not contained in any bounded region in strainspace (see Fig. 2).

For weak martensitic transformations the invariance of the energy (2) canbe limited to a finite subgroup of G due to the group-subgroup relation; corre-spondingly, the domain of the energy can be restricted to a neighbourhood ofthe reference configuration. This neighbourhood does not contain any lattice-invariant shears and only contains a finite number of energy wells. Such domainsare called Ericksen-Pitteri neighbourhoods (EPNs)24,25, see Fig. 2. Thereforethe classical framework of Landau theory and nonlinear elasticity24,26,4,23,13 ap-plies in these cases.

For reconstructive martensitic transformations, on the other hand, the sym-metry decreases along the transformation path, but increases again at the finalstate, and there is no reference configuration whose lattice group contains thoseof the two given phases.

We establish the following mathematical fact: the lattice groups of the twophases in any reconstructive transformation necessarily generate unbounded dis-tortions. This implies that in this case the barrier between the infinitely manyenergy wells of the crystal is at most equal to that of the underlying transfor-mation. Consequently no reduction to a finite number of wells in a boundedregion is possible (no EPN can be extracted). In particular, since one can findtwo states of the crystal related through an arbitrarily large distortion whichare only separated by a barrier as high (low) as the transition’s, such materialcannot resist certain arbitrarily large distortions; this creates defects in the lat-tice and makes the reconstructive phase change necessarily non-thermoelasticand irreversible. Further, no classical approach of the Landau type is possible.(Some authors have extended the Landau framework to encompass reconstruc-tive transformations by introducing a ‘transcendental order parameter’4, whichgives a partial description of the lattice periodicity characterized by the globalgroup G.)

We show in the methods section that in a reconstructive transformation anunbounded element of G is always generated. Precisely, one necessarily obtainsan element with an infinite period (i.e., whose powers can become arbitrarilylarge), akin to slip and twinning in the parent phase. We further notice thatthe most common reconstructive transformations involve phases with maximalpoint symmetry, i.e., the primitive-cubic, the fcc, the bcc, or the hexagonalsubgroups of G. One can show that, when suitable lattice bases are considered,the phase changes arising in these cases generate the entire group G, with its fullset of lattice-invariant shears. We elaborate on the fcc-to-bcc transformationbelow.

The impossibility to restrict the symmetry to a finite subgroup of G, and theenergy domain to a suitable EPN, has dramatic implications for the variationaltreatment of reconstructive phase transformations. Indeed, if one assumes thatthe deformation at each time is determined by minimizing the free energy subjectto external forces and boundary conditions, the invariance of the energy underthe whole group G implies that the solid cannot resist any shear27. In practice,dynamics and defects, including dislocations, moderate this phenomenon, whichwe revisit through our numerical simulations.

We now focus on a concrete case-study, demonstrating that the compositionof an fcc-to-bcc (forward) transformation and a bcc-to-fcc (reverse) transforma-tion can in fact result in a lattice-invariant shear. We start from an fcc lattice

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a

b

Figure 2. Schematic representation of weak vs. reconstructive transforma-tions in the space of lattices. a Weak: a uniaxial deformation (dashed blackarrows) of the fcc lattice (represented by squares) can give three equivalent bctlattices (circles). The reverse transformation (dashed green arrows) returns tothe original fcc configuration. All the transformation strains are confined withina single Ericksen-Pitteri neighbourhood (EPN), such as the dashed circle. b Re-constructive fcc-to-bcc: the transformation leads from an fcc to three equivalentbcc lattices, and the reverse transformation from each bcc can proceed to threedistinct but equivalent fcc lattices, and so on. No EPN can be singled out.

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aligned with the coordinate axes, and subject it to a uniaxial stretch Uz alongthe z axis:

U (λ)z =

1 0 00 1 00 0 1/λ

,

neglecting volumetric changes. For λ between 1 and√

2, the product lattice hastetragonal symmetry, and we have a weak transformation. When λ is equal to√

2, we obtain the Bain stretch U(√

2)z = Bz which produces the bcc lattice in

a reconstructive transformation, see Fig. 3. Now consider the reverse bcc-to-fcctransformation: it can occur in three possible ways, by the symmetry of the bccphase. A crucial point is that the symmetry axes of the bcc lattice are differentfrom those of the initial fcc crystal. One of the three possibilities for the reversestretch corresponds to B−1

z , and leads back to the original lattice. The other twoare stretches along the (1,±1, 0) directions (still using the original coordinatesystem). For instance, let us consider the reverse stretch V along the (1, 1, 0)fccdirection, which can be expressed as

V = QB−1z QT ,

where Q ∈ SO(3) is the 90-degree rotation with axis (1,−1, 0), which belongsto the symmetry group of the bcc phase. After applying V , the final lattice isagain fcc; however, the total transformation gives

V Bz = QB−1z QT Bz = RS,

where R is a rotation and S is a shear on a {111}fcc plane along a 〈112〉fccdirection. One can check that S brings the original fcc basis to a G-equivalentone, that is, the transformation V Bz restores the fcc lattice to itself up to theinessential rigid-body rotation R (see Fig. 3). Successive iterations of V Bz thengenerate larger and larger lattice-invariant deformations, and similar strategiescan generate the entire group G. The barrier between the G-related shearedconfigurations originated in this way is exactly equal to the barrier of the phasetransition. One can check that if only part of the crystal undergoes the V Bz

transformation, while the rest goes back to the original state, a Shockley partialdislocation with Burgers vector 1

6 〈112〉 is formed28. The double transformation(V Bz)2 then gives the Burgers vector 1

2 〈110〉. Both kinds are typical of fcccrystals. When starting from a bcc crystal, this same mechanism generatesdislocations with Burgers vector 〈111〉bcc on {112}bcc planes, which are amongthe most common ones in bcc lattices.

We now illustrate the phenomena discussed above for finite domains, alsoin the presence of boundary conditions, with numerical simulations done forsimplicity for a square-to-hexagonal (s-h) transformation in two dimensions.We consider a 50 × 50 grid of atoms interacting with a three-body nearest-neighbor potential29, which produces for the crystal a two-dimensional versionof the G-invariant energy in (2).

The simulation of a shearing experiment of a crystal close to a reconstructivephase transition is shown in the left panel of Fig. 4. As the left and the rightboundaries are progressively displaced, dislocations form in the lattice, whichlead to irreversible plastic deformations and defects in the crystal.

In contrast, the right panel of Fig. 4 shows the same simulation for a crystalclose to a square-to-rhombic (s-r) symmetry-breaking (weak) transition. In this

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a

b

c

Figure 3. Shear generated for a face-centered-cubic (fcc) to body-centered-cubic (bcc) transition, shown in a fcc-bcc-fcc cycle. a An fcc lattice, with body-centred-tetragonal (bct) cell in red. b The uniaxial Bain stretch Bz transformsthe bct cell to the bcc one. c The inverse Bain stretch V = QB−1

z QT transformsthe crystal back to fcc (blue fcc cell highlighted), but sheared relative to theoriginal. The green arrows show a basis undergoing the lattice-invariant shear.

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a c

b d

Figure 4. Reconstructive transformations (left panel) generate dislocations,weak ones (right panel) do not. a Section of the energy profile of a crystal closeto a square-to-hexagonal (s-h) transformation, along a s-h-s line. The energyhas the full invariance (2), with minima s and h plotted respectively at positions0, 1, and 0.5, 1.5, etc. b An incremental shear test with boundary conditions onthe left and right side results in dislocations. c Section of the energy profile ofa crystal close to a square-to-rhombic (s-r) transformation, along a s-r-h line.Here the square states s in 0, 1, etc. are metastable, and additional minima arepresent at intermediate rhombic configurations r, with hexagonal maxima at0.5, 1.5. etc. d The same incremental shear test results in no dislocations, butonly reversible transformation twins, because starting from the square minimumin 0, the ‘far-away’ square in 1 is not reachable by overcoming a small energybarrier.

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case, no dislocations arise in the lattice. Instead, typical layered martensitictwins mixing with the higher symmetry parent phase can be observed, whichaccommodate the imposed boundary condition in a reversible way (this is atthe origin of the memory effect). These observations confirm the importance ofthe symmetry of the stable states of a crystal for determining the macroscopicreversibility properties of the phase transition.

Methods

To prove that in a reconstructive transformation an aperiodic element of G(called GL(3, Z) in algebra) is generated, we first notice that any lattice groupis finite, and conversely any finite subgroup of G is included in the lattice groupof some lattice (see e.g. Proposition 3.5 in Pitteri and Zanzotto23). Thus atransformation is weak if and only if the lattice groups of the two crystal phasesgenerate a finite group. Therefore a reconstructive transformation producesan infinite subgroup of G with a finite number of generators. Such a groupnecessarily contains an element with no finite period as a consequence of theBurnside-Schur Theorem on periodic groups.

We finally establish that for suitable lattice bases, any reconstructive trans-formation involving Bravais lattices with maximal symmetry will generate theentire group G. Indeed, it is readily verified that for suitable pairs of subgroupsin G belonging to the four arithmetic classes with maximal point symmetry(i.e. the hexagonal and the primitive, face-centered, and body-centered cubicclasses) one can indeed produce all the generators of G, i. e. a suitable reflection,permutation, and simple shear30.

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Acknowledgements

The work of S. C. was partially supported by the DFG SchwerpunktprogrammAnalysis, Modeling and Simulation of Multiscale Problems. G. Z. acknowledgesthe partial support of the Italian Cofin. MURST2002 Modelli Matematici peri Materiali. S. C. and G. Z. acknowledge the partial support of the EU TMRnetwork Phase Transitions in Crystalline Solids. K. B. and J. Z. acknowledgethe partial financial support of US AFOSR and the US ONR.

Competing interests statement

The authors declare that they have no competing financial interests. Correspon-dence and requests for materials should be addressed to K. B. (bhatta@caltech.edu).

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